Well-Behaved Versus ”Non-behaved”models of Growth:

The Economics of Alice in Wonderland

Martin Kaae Jensen∗

April 2003

Abstract

Can an increase in the rate of savings slow down long-run growth ?

Can a tax on capital lead to higher growth even though the revenue

is not spend on R&D, or for some other productive purpose ? Is

a perfectly competitive, non-distorted economy safeguarded against

’growth traps’, implying that ’free market’ development strategies al-

ways work ? The present study shows how any growth model with

an infinite horizon consumer can be reduced to a so-called LI-LS di-

agram, leading to the answering of these and related questions. The

answers are respectively: yes, yes, and no.

Keywords: New growth theory, Savings, Growth trap, Taxes, Multiple Bal-anced Growth Equilibria, Cambridge controversy.JEL-classification: E2, O1, O3, O4.* I would like to thank Carl-Johan Dalgaard whose comments have influencedthe present paper significantly. Thanks are also due to Philippe Aghion, An-tonio Ciccone, Birgit Grodal, Neri Salvadori, Manuel Santos, and AntoniaSwann all of whom have influenced the paper by comments and suggestions.Financial support from the Danish Social Research Council is gratefully ac-knowledged by the author.Correspondence: Martin Kaae Jensen, Brown University, Department of Eco-

nomics, 64 Waterman Street, Providence, RI 02912, US. E-mail:

Martin Jensen@Brown.edu. Homepage: http://www.econ.brown.edu/fac/

Martin Jensen

0

1

I. Introduction

”...the simple tale told by Jevons, Bohm-Bawerk, Wicksell, and otherneo-classical writers - alleging that, as the interest rate falls in consequenceof abstention from present consumption in favor of future, technology mustbecome in some sense more ’roundabout’, more ’mechanized’, and ’more pro-ductive’ - cannot be universally valid.” (Samuelson (1966) p.568).

Economics has its conventional wisdom: If a tax is placed on capital, thisleads to lower growth rates, unless growth is purely exogenously driven orthe revenue is spend at some productive purpose. If an economy is caughtin a ’growth trap’, this is ultimately due to some sort of distortion or imper-fection, implying that if the unfortunate economy could be tranformed intoa perfectly competitive one, accumulation of wealth would follow. Finally,if consumers for some reason become more thrifty and increase their rate ofsavings, the long-run growth rate will increase.The present paper’s main purpose is to show that the previous conclusions

are not in any way universally valid. In fact they are intimately tied toan implicit assumption, named well-behavedness, which is present in almostevery growth model in the literature.Figure I depicts the long-run diagram of a model in the Romer (1990)

tradition.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

rate of interest

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4factor of growth

Figure I: Long-run Diagram

The increasing curve in the figure is a long-run savings (LS) curve. Givensome level of the interest rate, the LS curve associates a rate of growth

2

in consumers’ savings. This curve is unambiguously increasing under stan-dard assumptions: A higher rate of interest will lead the consumers to post-pone consumption, hence will increase relative intertemporal savings in theeconomy.1 The decreasing curve is a long-run investment (LI) curve. If thisis non-increasing - if a higher rate of interest does not make firms increasetheir rate of accumulation - these curves will intersect at most once: at theunique balanced growth equilibrium (BGE) of the economy.In section II it is shown that long-run diagrams can be derived and studied

for every single model in the literature which has a Ramsey-type aggregateagent at the consumption side. In fact, even the most exotic productionset-up with thousands of capital goods, externalities, and whatever else, willreduce to a long-run investment curve. And if attention is devoted to bal-anced growth, these curves embody all the information needed to answercomparative long-run questions such as: What is the effect of a higher taxrate on the growth rate ?A decreasing long-run investment curve is representative of most new

growth models, including those of Romer [1990], Grossman-Helpman [1991],and Aghion-Howitt [1992]. If a model is of the ’AK’ variety (Frankel [1962],Romer [1986], Lucas [1988], and Rebelo [1991]), then the LI curve will behorizontal. Finally, if growth is exogenous, the LI curve will be a vertical lineat the exogenous rate of growth. What these situations have in common isthis: The LI curve is non-decreasing over the interval of growth rates whereit is defined. This is what we name well-behavedness. Intuitively a LI curve iswell-behaved if a higher rate of interest never makes firms more ’productive’,i.e., makes them increase their rate of accumulation.Now imagine that an LI curve is not well-behaved (non-behaved, hence-

forth). Figure II shows two types of non-behaved LI curves (’type A’ and’type B’, say), together with (increasing) LS curves. A tempting conjecture isthat such outcomes require ”perverse” conditions; wild distortions, concave-convex productions sets, or some other non-standard device. After all, anincreasing LI curve means that when firms face a higher rate of interest (costto capital), they respond by increasing their rate of accumulation. A cen-tral contribution of the present paper is intented to shoot this apparentlyself-evident statement down, by providing an example of a convex, perfectly

1In the special case of linear utility, the LS curve will be horizontal: For a fixed rate ofinterest, consumers will be willing to accumulate wealth at any feasible pace.

3

2.62

2.64

2.66

2.68

2.7

2.72

int.rate,%

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6rate of growth, %

30

40

50

60

70

int.rate,%

4 5 6 7rate of growth, %

Fig.IIA Fig.IIB

Figure II: ’Non-behaved’ long-run investment curves

competitive (CPC) economy with no externalities or other distortions, givingrise to a ’type B’ LI curve. Like the LI curve in the figure, the examplegives rise to three BGEs (two of which are stable). Imagine that intersectionpoints on the increasing part of the LI curve are stable BGEs. In the exam-ple this is actually the case for certain parameter values. Now move the LScurve downwards (the effect is most easily seen if the LS curve is taken to behorizontal in figure IIB). Such movements correspond to increasing savingsrates, or, decreases in capital taxation. The first effect will be a decrease ininterest rates; consumers supply more capital hence the cost to investmentsgo down. But when the interest rate falls and the economy is at an increas-ing segment of the LI curve, this does not make the growth rate increase -it makes the firms’ rate of accumulation decrease. In balanced growth equi-librium the effect will be negative: A higher rate of savings will lead to alower rate of growth. Needless to say, had the LI curve been well-behaved(non-increasing), the conventional wisdom would have between correct: thegrowth rate would unambigiously have increased.All of this would have been easier to dismiss if it was not because em-

pirically very little seems to support the well-behavedness assumption. Theconnection between taxes and growh is muddy at best (cf. Uhlig and Yana-gawa (1996)). As for the savings rate and growth, this is an old debatedating back at least to the days of Keynes’ General Theory. Here, again,emirical evidence is mixed (Carroll and Weil (1993), Rodrik (1998)). To thebest of knowledge, the non-behaved example in this paper is the first which

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shows that the savings rate may very well be negatively correlated with thegrowth rate. The case studied by Uhlig and Yanagawa (1996), which alsopredicts that taxes and growth may be positively correlated, is very differentfrom non-behavedness as defined here. There the explanation is that in anOLG-a-la-Diamond set-up, savings stems from labor income alone. Hence atax which transfers from capital to labor is likely to increase savings, hencegrowth. The present example can be read as saying that even if taxes dotend to retard savings, the intention of savers - to reallocate towards presentconsumption - may ”backfire”, because firms choose to raise their rate ofaccumulation in the face of a higher (after-tax) interest rate.

II. The LI-LS Approach

The purpose of this section is to show that, despite new growth theory’sdismissal of aggregate production functions, it still at the very core adheresto ’well-behavedness’ assumptions which parallel those leading to a Clarkiansurrogate production function in the sense of Samuelson [1962]. These as-sumptions drive virtually all bottom line conclusions, hence are far from beinginnocent. From a more constructive perspective, the LI − LS approach ofthis section provides a unifying and extremely simple way of studying growthmodels, hence it can be used to analyze many other questions than the onesfocused upon in this paper.

The Long-run Savings Curve

Consider a discrete-time, infinite horizon economy with a representative con-sumer, and a single consumption good. Consumption sequences are evaluatedaccording to a standard time-separable CES utility function:

U(c0, c1, . . .) =∞

t=0

δtc1−αt

1− α, α ≥ 0 .(1)

The intertemporal elasticity of substitution is σ = α−1 ∈ R++ ∪ +∞.In the limit case where α → 1, (1) reduces to the Cobb-Douglas form:U(c0, c1, · · ·) = t δ

tlog(ct).

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Denote net discounted income at date 0 by W > 0. This includes thediscounted stream of labor income, and the value of the stock of resources heldat date 0. A price sequence for the consumption good is denoted (p0, p1, . . .),pt > 0. The objective of the consumer is to find the consumption sequencewhich maximizes (1) and satisfies:

∞

t=0

ptct ≤ W(2)

A solution will be interior, and (2) hold with equality. Hence when α > 0the following sequence of Euler conditions must be satisfied (we return to thecase where α equals zero below):

ηpt = δtc−αt , t = 0, 1, 2, . . .(3)

where η > 0 is the Lagrange multiplier. The implicit interest rate on theconsumption good is defined by: rt =

pt−1pt−1. When this is constant, rt = r,

for all t, define:

s(r) = δ1α (1 + r)

1−αα ,(4)

It may be checked that s(r) is in fact the rate of savings at every datewhen r is the ruling rate of interest.2 From now on only those r > 0 forwhich 0 < s(r) < 1 will be considered. This requirement will be met undersuitable ’cross-restrictions’ on the consumption and production sides (suchassumptions are necessary for the existence of an equilibrium, cf. Jensen[2002b]).Denote savings at date t by St > 0. Since we have discounted all income

to date 0,3 income at date t derives entirely from savings at date t− 1, St−1.At the interest rate r, this produces income (1 + r)St−1 at date t. It followsthat savings at date t will be,

St = s(r)(1 + r)St−1 ,(5)

2The savings rate at date T is: sT =∞t=T

ptct− pT cT∞t=T

ptct. Normalize η to 1 in (3), isolate

ct, and insert to get: sT =∞t=T

δtα p

α−1α

t − δTα p

α−1α

T

∞t=T

δtα p

α−1α

t

. Then insert pt = (11+r )

tp0, and reduce

to (4).3It is well known that this forward market formulation is equivalent to the sequential

market formulation under a transversality condition.

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since s(r) is the rate of savings.To arrive at the long-run savings (LS) curve, define gs =

St−St−1St−1

, the rate

of growth in savings, and insert (4) in (5) to get:

r = δ−1(1 + gs)α − 1(6)

If gs is replaced with gc =ct−ct−1ct−1

, the growth rate of consumption, equa-

tion (6) turns into the familiar Keynes-Ramsey rule of optimal resource al-location. This is not surprising since in a balanced growth equilibrium thegrowth rate in savings and consumption will be the same: gs = gc.In figure 1 the LS curve is depicted for the parameter values α = 0.5

and δ = 0.53 (the increasing curve). The LS curve may be convex (α > 1),concave (α < 1), or linear (α = 1). If utility is linear (α = 0), the LS curvewill be horizontal at r = δ−1. Note that the LS curve will always be non-decreasing (and increasing unless α = 0). Economically this is simply due tothe fact that the savings rate, s(r), is non-decreasing in r (cf. equation (4)above).

Long-run Investment Curves

Imagine a production sector which produces a single consumption good bymeans of M + 1 factors, where M is the number of human and physicalcapital goods, and the last good is a primary resource (labor). Take a levelof the interest rate, r > 0, as given, and consider a corrsponding steady state(if one exists). By definition all capital goods’ prices decrease at the sameconstant rate, r

1+r, so all that is needed to characterize the capital price path,

pt =1

(1+r)tp0, is a relative price vector at date 0, p0 ∈ RM

+ . Now assume thatthere is ”constant returns to reproducible factors”, i.e., for a given vectorof primary resources, the aggregate production set is a convex cone in theset of reproducible goods (the capital goods and consumption). Assume thatprimary resources are in fixed supply. If firms maximize profits by choosing aproduction sequence such that the input of all capital goods grow at the sameconstant rate, g,it must then hold that the real wage grows at this same rate- otherwise profits would be either increasing or decreasing and the economycould not be in equilibrium.4 If finally it is assumed that the factor markets

4Of course it is an assumption that firms cannot earn such unbounded profits, but itis a reasonable one, clearly satisfied for all models in the literature.

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clear at all dates, then this system closes: For a given rate of interest thereexists a list of growth rates (perhaps only one) such that clearing of the factormarkets and profit maximization implies that the real wage and the capitalgoods expand at one of these growth rates. For each growth rate, there is acorresponding relative price vector which clears the sequence of markets atevery date. What is more: When consumers are assumed to maximize utility,Walras’ law holds. Hence, when all factor markets clear, the consumptionmarket will automatically clear at all dates since in a steady state a marketclears if and only if it clears at just one date.Letting the interest, r, vary, the previous construction yields a curve: the

long-run investment (LI) curve. Given some level of interest, r > 0, a setof corresponding growth rates can be calculated all of which are supportedby profit maximizing firms whose factor markets clear. The LI curve can begiven a perfectly rigorous derivation, but here I shall try to ”convince” thereader by a number of examples in stead.5

11

11.5

12

12.5

13

rate of interest, %

0 1 2 3 4 5 6rate of growth, %

Figure III: LI-LS curves in the ’AK’ model

A = 1.12, α = 0.7, δ = 0.91

To take the simplest possible example, assume that there is only onegood in the economy (a combined consumption-investment good), and anaggregate firm with the following production function:

yt = Akt−1 .(7)

5The construction and proofs can be found at www.econ.brown.edu/fac/Martin Jensen.

8

Here yt denotes output of the good at date t, kt−1 input at date t−1. This isoften referred to as an ’AK’ model (see Frankel [1962] and references therein).The firm will be profit maximizing at date t − 1 if it chooses kt−1 ≥ 0 suchas to solve:

maxkt−1≥0

ptAkt−1 − pt−1kt−1(8)

This yields the following necessary and sufficient condition for a non-trivialsolution (i.e. a solution where kt−1 > 0):

A =pt−1pt

= 1 + r .(9)

Hence, in equilibrium, the rate of interest must be constant and equal toA − 1. There is no separate factor market which must clear, and there isno primary resource. So the long-run investment curve trivially exists andis horizontal at r = A − 1. Economically, this mirrors the case with linearutility mentioned above, now at the production side: With linear productionfunctions, firms will be willing to accumulate at any pace at a fixed level ofthe interest rate. If the interest rate is below this level, firms earn infiniteprofits, if it is above this level, firms close down. Both are in disagreementwith the existence of equilibrium. Note also that A cannot assume all possiblevalues: It must be such that s(r) = s(A − 1), the consumers’ savings rate,lays in the interval between 0 and 1. Comparing with (4) this will be the

case if and only if 0 < (δA1−α)1α < 1. In figure II the long-run investment

curve with A = 1.12 is depicted together with the LS curve for α = 0.7 andδ = 0.91.Now this example is admitably special. In particular, the rate of interest

will be constant at any equilibrium trajectory, and there is instant initialadjustment to the BGE (cf. Rebelo [1991]). That the production side deter-mines the interest rate independently of the consumption side is characteristicfor a large class of models, however. In fact this description of an economydates back to the classical economists and is explicit in for example Ricardo’ssystems of simple reproduction. The Leontief model (Leontief [1928]), thevon Neumann model under non-joint production (von Neumann (1937), Mor-ishima (1961)), and more recently Romer [1986] and Lucas [1988], all lead tohorizontal LI curves.

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Next consider the neo-classical growth model where the aggregate firmhas a production function of the Cobb-Douglas form:

yt = Bkβt−1(AtL)

1−β, 0 < β < 1, B > 1(10)

L > 0 is the constant supply of labor and At a technological trend growing ata constant exogenous rate: At = (1+gA)

tA0. In this case profit maximizationwith respect to kt−1 yields the first-order condition:

Bβ(LAtkt−1

)1−β =pt−1pt

(11)

The long-run investment curve describes the rate of growth in investmentswhich the firms will persue at a given rate of interest. Setting pt−1

pt= 1 + r

we see that if the firm faces a constant interest rate, the only possible choicefor the firm is a capital sequence of the type kt = (1 + gA)

tk, k > 0 (sinceAt = (1 + gA)

tA0). As regards the LI curve this will therefore be verticalat the exogenous factor of growth 1 + gA. Regardless of the rate of interest,firms will invest such as to follow the exogenous technological growth trend,i.e., gI = gA, where gI =

kt−kt−1kt−1

is the growth rate of investments. Thefirms will do so by adjusting the capital-labor ratio of the technique in useto equalize the marginal product of capital with the interest rate. As for thereal wage, note that profits is Bkβt−1(AtL)

1−β − rkt − wtL. Clearly, and asmentioned this is a general fact, wt = (1 + gA)

tw0, otherwise profits couldnot be 0 at all dates. In contrast with the AK model above, the presentmodel will only by coincidence begin exactly on the balanced growth path.It will, however, converge to it (Cass [1965]). Also, the interest rate is notdetermined independently of the consumption side, but as the value whichequalizes the growth rate of savings and investments.Having now seen examples of a horizontal and a vertical long-run invest-

ment curve, let us finally consider the intermediate case where the LI curveis decreasing. A number of new growth models give rise to this outcome suchas the creative destruction model of Aghion and Howitt [1992], and the qual-ity ladder model of Grossman and Helpman [1991]. Here we shall considerthe increasing variety model of Romer [1990]. That model is formulated incontinuous time, which, however, does not make a significant difference as faras LI-LS curves are concerned. The following equation generally describesthe LS curve in continuous time (gs ≡ ∂S(t)/∂t

S(t)):

10

0.4

0.6

0.8

1

1.2

1.4

1.6

rate of interest

0.6 0.8 1 1.2 1.4 1.6factor of growth

Figure IV: LI-LS curves in the ’old’ growth model, 1 + gA = 1.03

r = σgs + ρ(12)

Where σ is the intertemporal elasticity of substitution, and ρ the rate of timepreference. As for the production sector, the following equation describes thelong-run investment curve in the Romer model (Romer [1990, p.92]):

r = χH − ΩgI(13)

where gI ≡ ∂I(t)/∂tI(t)

. χ, Ω, and H are positive constants determined by con-ditions of production including factor supply - hence they are independentof the consumption side (for example, H is the level of skilled labor in theeconomy). In this example the LI-LS curves are linear, the LS curve is in-creasing (as usual), and the long-run investment curve is decreasing. Therate of growth of investments will depend on the interest rate: The lower theinterest rate, the faster the rate of growth in firms’ investments. In contrastto the previous two cases, neither the growth or interest rates are determinedby one sector independently of the other.6 Also clearing of the factor marketsfor human capital, intermediate goods, and patents, are needed to derive theLI curve. This leads to a distinguishing feature, namely that the LI curvedepends on H, the level of skilled labor. This is a so-called scale-effect (seeDalgaard and Kreiner [2001] for a throughout discussion). Generally, the LI

6The exception being the linear utility case leading to a horizontal LS curve, hencedetermining the interest rate from the consumption side alone.

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0.8

1

1.2

1.4

1.6

1.8

2

rate of interest, %

0.5 1 1.5 2 2.5rate of growth, %

Figure V: LI-LS curves in the Romer [1990] model

σ = ρ = 0.5, χ = 0.7, H = 3,Ω = 0.3

curve could depend on a number of variables including those related to pro-ductivity, factor supply, sector composition, and the degree of competition.An exogenous change in any one of these will shift the LI curve, producingan increase, a decrease, or neither of these, in the growth and interest rates.

Well-behavedness

The examples considered so far cover a wide array of different models andone could go on for years on end computing the LI-LS curves of models in theliterature. In fact, shown formally in the working paper version of this pa-per, any production side which supports unbounded growth reduces to an LIcurve. Regardless of the type of competition, presense of externalitites, num-ber of capital goods, etc., etc., there is always an LI curve lurking behindthe scenes - and driving the conclusions. When one studies the objectivecharacteristics of existing models, they may seem very different (and in asense they are of course). But what one soon realizes is that essentially all ofthem share a feature which ”happens” to be the crucial one. Just as Samuel-son’s ”surrogate production function” (Samuelson (1962)) summarizes therelationship between interest and wages holding in stationary states, the LIcurve summarizes the relationship between interest and growth rates in abalanced growth equilibrium. Like Samuelson’s concept of a ”Clarkian para-ble” posits a monotone (inverse) relationship between wages and interest; the

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previous models’ LI curves share a ”pseudo-Clarkian” feature which I havechosen to call well-behavedness. Indeed, considering the previous models itis seen that the long-run investment curves are all non-increasing over theinterval where gI is defined. This of course includes the ’old’ growth modelexample where gI is only defined in a single point. Since the LS curve is al-ways non-decreasing, it is clear that it will intersect with the LI curve at mostonce when well-behavedness is assumed. Hence for one thing, a well-behavedLI curve implies that there is a unique balanced growth equilibrium. In theintroduction it was explained that uniqueness is only the most obvious im-plication, however. To be more precise at this point, consider a proportionaltax, τ ∈ [0, 1), on interest income, which leads to the following modified LScurve (r is the gross interest rate, (1− τ)r, the net interest rate):

r =δ−1(1 + gs)α − 1

1− τ(14)

If τ is increased, the after tax interest rate decreases, which shifts the LScurve upwards. This will have no effect on the gross interest rate but maximaleffect on the growth rate if the LI curve is horizontal. More generally, it isseen that the growth effect will unambigiously be negative if the LI curveis decreasing (in the exogenous growth model there is, of course, no effecton the growth rate). The same conclusion holds if we were to consider anexogenous parameter change which lowers the savings rate for all levels ofthe interest rate (for example: a decrease in the ”patience parameter”, δ).Again, if growth is endogenous and the LI curve is well-behaved, this willunambigiously make the long-run growth rate decline. Actually, it is quiteclear that well-behavedness makes comparative dynamic exercises incapableof suprising us; thus the conventionality of conventional wisdom - we knowall the conclusions before making the calculations.There are several reasons why this motivates further discussion. First,

there is not much which empirically supports well-behavedness, at least as auniversal feature. Thus it is well known that the connection between taxesand growth is ambigious; some would even say that it is positive (cf. Uhligand Yanagawa (1996)). As for the savings rate and growth, this is a classicaldebate dating back at least to the days of Keynes’ General Theory. Here,again, emirical evidence is mixed (Carroll and Weil (1993), Rodrik (1998)).Once confronted with such facts, one’s first responce is to think that

some sort of imperfection or distortion is at work. This is, of course, because

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the intuition which underlays a well-behaved LI curve is so grounded in usthat a departure from its implications appear worrysome. It is then as if aninner drive makes us conclude that ”something is wrong”. Now, where anabstract perspective as that of LI-LS curves has something to contribute tomacroeconomic modelling, is when it is able to point out that it is intuitionwhich may be wrong. This is the objective of the next section.

III. ’Non-behaved’ Long-run Investment

Curves

In the last section was shown how growth models can be framed in a waywhich is close in spirit to AD-AS diagrams of Keynesian income models. Asin that framework, the LI-LS construction places focus upon the shape of therespective curves, in particular their curvature and monotonicity properties.When the LI curve has an increasing interval - when it is not well-behaved(non-behaved) - any one of the conclusions that follow from well-behavednessare subject to violation. Since the conventional wisdom of growth theory andwell-behavedness are essentially one and the same thing, an example of an LIcurve which is not well-behaved (non-behaved) is the obvious way to proceedwith the discussion.

Consider a perfectly competitive economy with two capital goods: physi-cal and human capital. The physical capital good is a perfect substitute forthe consumption good, c, so at date t the first market clears provided that:

ct + kot = y

ot(15)

kot is the input of ordinary capital at date t, yot the output. The second good

(human capital) is not consumed, so market clearing simply reads:

kht = yht(16)

where kht denotes input and yht output of human capital. Technology is time-

stationary, and a vector of outputs (yot , yht ) can be produced at date t by

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use of a vector of inputs (kot−1, kht−1) at date t − 1. Let Y ⊂ R4

+ denote thetransformation possibility set. Then an input-output vector is feasible if andonly if (kot−1, k

ht−1, y

ot , y

ht ) ∈ Y . To fix notation, we briefly consider the case

where Y is any closed convex cone.The production sector will maximize profits by solving the following prob-

lem at all dates, t = 1, 2, . . .:

max ptyot + p

ht yht − pt−1kot−1 − pht−1kht−1(17)

s.t. (kot−1, kht−1, y

ot , y

ht ) ∈ Y

Here pt > 0 (pht > 0) denotes the price of the first (second) good atdate t. At date 0 the consumer holds an exogenously given vector of goods(yo0, y

h0 ) 0, which is though of as produced by means of inputs used in

the past. The (aggregate) firm earns zero profit (constant returns) and theconsumers’ discounted income,W , consequently stems from selling the initialvector at date 0: W = p0y

o0 + p

h0yh0 .

In a BGE, the price sequences induce an overall rate of interest, r:(pt, p

ht ) = (

11+r)t(p0, p

h0). The initial price vector is subject to a normalization,

and here we shall take p0 = 1. It follows that in a BGE (17) reads:

max ( 11+r)tyot + (

11+r)tph0y

ht − ( 1

1+r)t−1kot−1 − ( 1

1+r)t−1ph0k

ht−1(18)

s.t. (kot−1, kht−1, y

ot , y

ht ) ∈ Y

Now consider (18) at t = 0:

max yo0 + ph0yh0 − (1 + r)ko−1 − (1 + r)ph0kh−1(19)

s.t. (ko−1, kh−1, y

o0, y

h0 ) ∈ Y

If (ko−1, kh−1, y

o0, y

h0 ) solves (19), then so does (1 + gI)

t(ko−1, kh−1, y

o0, y

h0 ), all

gI ≥ 0, t ≥ 0 (Y is a cone). Since Y is also convex, it follows that (19)is a necessary and sufficient condition for the balanced production sequence(1+gI)

t(ko−1, kh−1, y

o0, y

h0 ), t = 1, 2, 3, . . . to maximize the firms’ problem given

the price sequence pt = (11+r)tp0. Now, if the (aggregate) firm chooses such a

balanced production sequence, the human capital market will clear providedthat:

(1 + gI)tyh0 = (1 + gI)

t+1kh−1, t = 0, 1, 2, . . .(20)

which is satisfied for all t if and only if:

yh0 = (1 + gI)kh−1(21)

15

This in turn implies thatyh0kh−1

= 1 + gI , i.e., human capital’s output-input

ratio equals the investment growth factor. We return to the economic contentof this observation in a moment. From (21) follows that clearing of thesecond market together with (19) completely determine the growth rate ofinvestments, given the relative price ph0 > 0 and the rate of interest r > 0.Moreover, since r determines ph by the requirement that the firms’ problemhas a solution, r alone determines gI , so in fact (19) and (21) determine thelong-run investment curve. Of course this is valid for any production set Y ,a concrete example following next.Assume that the economy consists of two types of firms, each of which

has access to a Leontief activity (fixed coefficient production set). A firm oftype A can choose plans, (kot−1, k

ht−1, y

ot , y

ht ) ∈ Y A, where (kh is, without loss

of generality, normalized to unity):

Y A = (ko, kh, yo, yh) = λ(0.85, 1, 1.5, 1.04), λ ≥ 0(22)

A type B firm can produce according to:

Y B = (ko, kh, yo, yh) = λ(0.55, 1, 1.05, 1.07), λ ≥ 0(23)

The production side is subject to free entry of firms, so the aggregateproduction set, Y , is:

Y = (ko, kh, yo, yh) = αyA + (1− α)yB, yA ∈ Y A, yB ∈ Y B, α ∈ [0, 1]

Clearly Y is a closed, convex cone containing the origin. Intuitively, α ∈ [0, 1]is the fraction of firms of type A. Note that, if only one type of firms exist,yh will equal the investment growth factor 1 + gI due to the normalizationkh = 1 (see (21) above).For a given rate of interest, r > 0, there will be a set of relative prices,

ph, which is compatible with profit maximization. Once this is found it ispossible to determine the fraction of firms of each type that will be active.This will then determine the investment growth rate associated with r. Moreprecisely, considering (19), it is seen that a firm of type A will earn zero profitif and only if:

ph ≥ (1 + r)0.85− 1.51.04− (1 + r)(24)

16

If (24) holds with strict inequality, firms of type A will not operate sinceany production plan except (0, 0, 0, 0) yields negative profit. Likewise fortype B firms:

ph ≥ (1 + r)0.55− 1.051.07− (1 + r)(25)

The relationships (24) and (25) are depicted in figure VI. Since each firm is

0

1

2

3

4

5

p

1.2 1.3 1.4 1.5 1.61+r

Figure VI: (24) dotted, (25) solid.

inoperative to the right of its respective curve, the competitive equilibriumcombinations of r and ph will correspond to the outer envelope of the curvesin the figure. At the two intersection points both types of firms will generallybe operative, but everywhere else only one of the types will produce.It is immediately clear that the long-run investment curve of this example

will be non-behaved: For low rates of interest only type B firms will operate,hence the investment rate of growth will be 7%. Raising the interest ratewill eventually make firms of type B inoperative, and the growth rate falls to4% as firms of type A have taken over. This shift is well-behaved: A higherrate of interest slows down capital accumulation. However, as the interestrate further increases, the economy will switch back to type B firms andthe investment growth rate of 7%. Hence there is a critical level at whichan increased rate of interest leads to a higher rate of growth. Figure VIIdepicts the resulting LI curve. The horizontal line segments correspond tothe intersection points in figure VI, where firms of both types are active.At the vertical line segment the economy consists only of firms of one ofthe types. The LI curve and LS curve in the figure intersect three times.

17

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

int.rate

1.03 1.04 1.05 1.06 1.07 1.08growth factor

Figure VII: Non-behaved LI-LS curves, α = 4.86, δ = 0.896 in the LS curve

Hence this example gives rise to multiple BGEs: different initial ratios ofordinary to human capital lead to different long-run rates of growth.7 It ispossible to show that only BGEs on the vertical parts of the LI curve will belocally stable. So in the figure, the BGE ’in the middle’ is unstable, whilethe BGEs associated with the growth rates of 4 and 7 % are stable (it followsthat all firms will be of either type A or B on a stable BGE). This leadsto all the usual consequences of multiple stable steady states, for examplea temporary negative chock to the economy may have permanent negativeeffects. As a first conclusion therefore: Multiple (stable) BGEs is a definitepossibility in perfectly competitive market economies (with no externalities,non-convexities, or some other distortion). While the literature is by nowrich on examples of multiple BGEs (see e.g. Azariadis and Drazen (1990)),to the best of knowledge this example is the first one where multiplicity isnot associated with a convex-concave aggregate production set or some otherdeparture from standard conditions of perfect competition.The previous example also verifies a central claim of this study, namely

that the ’conventional wisdom’ is intimately tied to the well-behavednessassumption. Thus imagine that utility is linear (α = 0), and that a uniformproportional tax, τ ∈ [0, 1), is placed on savings income. In this case theLS curve will be horizontal at r = δ−1−1

1−τ (cf. equation (14) above). Assume

7As mentioned in the previous section, an intersection between the LI and LS curvesis not only necessary but sufficient for the existence of a BGE. Hence this claim is indeedvalid.

18

that δ = 0.76, and that the tax level is τ = 29%. Then the market rateof interest will be 44% and the (unique) BGE have a growth rate of 4%(compare with figure VII where the LS curve will be a horizontal line layingbelow the ’upper’ horizontal line of the LI curve). An increase in taxes toτ = 35% will make the interest rate increase to 49%, by shifting the LS curveupwards. This increase will make the growth rate increase to 7%, since allfirms of type A become inoperative in favor of type B firms. This feature -that a tax on capital may increase the long-run rate of growth for a uniqueand stable BGE - is striking to say the least. Clearly, this outcome wouldbe impossible if the LI curve were well-behaved: then a tax on capital wouldunambiguously lower the growth rate. By comparison with section II, thereader may also check that any change in δ or α which makes the savingsrate s(r) decrease, may lead to a higher growth rate by shifting the LS curveupwards. The reasoning is exactly the same as in the taxation example.Again, an upwards shift in the LS curve would lead to a lower rate of growthin all cases where the LI curve is well-behaved and not vertical.One could easily add more firm types (increasing the substitution possi-

bilities in the aggregate production set) such that the LI curve of the figureabove would eventually become smooth and have the shape of an inverteds (see figure II B in the introduction). I shall not pursue the details of thishere, though.The previous example has been deliberately chosen because it is simple

and shows the close relationship to the phenomenon of ”reswitching” (seeSamuelson (1966)) heatedly debated in the Cambridge controversy. Gener-ally speaking, a production plan reswitches if it becomes profit maximizingat non-connected intervals of the interest rate. Thus in the example above,firms of type B are most profitable both at low and high interest rates, butnot for interest rates in the middle zone - they reswitch. It is very impor-tant to make a destinction, however, because the present discussion is not arepetition of the Cambridge controversy. The LI curve is generally derivedassuming factor market clearing hence, technically, reswitching in the senseof the Cambridge controversy is not sufficient for non-behavedness. Con-versely, it is of no importance whether it is the same production plan thatrecurs (compare with figure IIA in the introduction), so non-behavednessis not sufficient for reswitching. The present discussion has a precise defini-tion of ”productive”, ”roundabout”, etc. (cf. the quote in the introduction),namely the relative growth in investments. The Cambridge controversy was

19

concerned with another (not always very precise) feature: the existence ofparable like aggregate production surfaces. To be sure the Romer (1990)model considered above would never lend itself to a ”surrogate productionfunction”, or a production function however defined (within reasonable lim-its), yet at the same time it yielded a well-behaved LI curve. Having madethis disclaimer, it is never the less clear that much of the spirit of the Cam-bridge controversy is present in the discussion of well-behavedness. Afterstudying hundreds of growth models with very different economic assump-tions, all of which end up affirming the conventional wisdom, it is easy toend up accepting this as universally valid. But, just as the Cambridge-UKpeople during the controversy, one can take a critical stand towards this wayof sneaking implicit assumption in through the ”educational backdoor”.Compared to the Cambridge controversy, the present discussion is much

more economic in nature, as the questions about taxes, savings rates, andgrowth testify to. The example above is - well, just an example, but muchmore economically intuitive examples could be constructed which lead to thesame conclusion: There is nothing universal about a well-behaved growthmodel; and thus there is nothing universal about the conventional wisdomof growth theory. Finally, the specific nature of production is not importantfor this observation, since any model can be reduced to an LI-LS diagram.

IV. Conclusions

The LI-LS diagram of section II is perhaps the simplest way of studyinggrowth models. Yet, the main advantage is its unifying nature. The frame-work disregards short-run dynamics, but brings the crucial long-run proper-ties into focus. Well-behavedness of the LI curve was thus shown to be theproperty underlying conventional wisdom: If the LI curve is well-behaved,taxes on capital or lower rates of savings will always lead to a lower rate ofgrowth. Well-behavedness also implies uniqueness of the BGE, hence rulesout ’underdevelopment traps’, and the possibility of affecting long-run growthby temporary policy measures.It has been the purpose of this paper to argue that something may be

wrong with the intuition which guides model builders in growth and devel-

20

opment economics. Take underdevelopment. If capital is mobile the interestrate will tend to equality across countries. An underdeveloped country has”low productivity”, and so the interest rate is high. Therefore capital willflow into the poor country, and productivity will increase untill the interestrate is at the level of the developed countries. True or not true ? True if theproduction sides are well-behaved. Not true if they are not. As figure II.A.in the introduction shows, in the face of a non-behaved LI curve the interestrate might actually be lower in the underdeveloped low-growth country, sothe question becomes: why doesn’t capital flow from the poor to the richcountries ? In fact it often does, so this is not the question but the answer.Taylor [1997] has examined the reward to direct investment in underdevel-oped countries relative to the US. And he finds just that: The reward toholding capital is lower in almost all underdeveloped countries than it is inthe US. If this is so, opening capital markets will not improve upon the un-derdeveloped country’s situation. Only free access for the country to sell itsoutput in developed countries will work. And the fact that almost all foreigndirect investment is horizontal (Brainard [1997]), i.e., aimed at selling theproducts in the same place as where the goods are produced, is sufficientevidence that this is far from being the way of the world.Evidently, the discussion has much in common with the Cambridge con-

troversy which raged some fifthy years ago. But as explained in section III,it not not the same debate. There need not exist a Clarkian parable surro-gate production function in the sense of Samuelson [1962] for the LI curveto be well-behaved. In fact, new growth theory has for the main part leftthe neo-classical production behind in favor of a disaggregated characteriza-tion of capital. The question is about expansibility, more precisely about thegrowth rate at which a chosen vector of capital goods expands when marketsclear. Non-behavedness, which has nothing perverse about it, means thatthere is no monotonic relationship between the interest and growth rates(while there may be one between factor prices): Although the interest rateincreases, relative prices may change in the process such that even thoughcapital goods are more expensive to finance, it is never the less profitable toextend their use. Intuitively, production processes which use more capitalinputs may become more profitable than the rest even though the interestrate increases, simply because they also produce more of an output for whichthe relative price increases along with the interest rate. The example in sec-tion IV showed that this phenomenon is not bound up on non-convexities,

21

externalities, or some other departure from the perfect market economy. Inparticular, the example shows that the previous description of underdevelop-ment is not tied to the fact that the underdeveloped economy is imperfect.Unlike other examples of growth traps in the literature, in the present casethere is no cure which will turn the economy into a perfectly competitive oneand offset growth. To the extend that a ”growth trap” constitutes a failureof an economic system, market economies may embody such failures.Perhaps the most surprising conclusion of this paper, is that an increase

in consumers’ thriftiness, or policies aimed at increasing savings, may endup slowing down growth. Say that people in Zimbabwe suddently becomemore thrifty. This will, in all likelihood, lower the interest rate which in itselfwould lead to faster accumulation. The problem is that relative prices willchange too. And it may be that the change is such that the most expansibleproduction technologies’ inputs become relatively more expensive. If thishappens, then the original intention (if there ever was one), will ”backfare” inthat the consumers eventually will end up consuming less than they otherwisewould have enjoyed. As a corollary, the people of Zimbabwe cannot a prioribe said to be less thrifty than, say, the people of the US. If we measurethe value of capital, K, and investments, I, and assume that K = I − δK,and S = I, where S is savings, the savings rate is equal to the productof the gross growth rate in capital ( K

K+ δ), and the capital-output ratio.

So, if the capital-output ratio is higher in the US the savings rate will haveto be (much) lower in Zimbabwe. If investment/savings and capital stockare measured in international prices (or US prices), this outcome is true byconstruction since this yields a higher capital-output ratio in the US thanin Zimbabwe. But what does it tell us about thrift and savings rates ?Absolutely nothing.8 If this story has any bearing, we would expect therelative price of investment goods to be (much) higher in Zimbabwe than inthe US so that the capital-output ratio measured in local prices tended to behigher in Zimbabwe than in the US. Empirically, this is well-known to be thecase. Thus Hsieh and Klenow (2002) find not only this but, to quote fromthe source: ”When measured in nominal terms (i.e., at national prices ratherthan at PPP prices), investment rates are little correlated with income.”.

8The reader should note that here there is a close connection to the capital controversy.If a tax on capital implies a relative price change, then what happens to the value of theexisting capital stock ? This is a good question, and the methods usually employed whencalculating capital stocks are completely incapable of taking account of it.

22

All said and done, this paper has only suggested that well-behavednessneed not characterize the real world. I have mentioned empirical evidencewhich is favoureble towards this stand, but more work would have to be doneto settle the question. Even though the findings in this project definentlysurprised me, maybe they do not surprise everybody. But it is my hope thatat least some readers will see the conclusions of this paper as a challengeworth taking up.

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